Let $P$ be an integer valued polynomial with degree $> 1$, is it a theorem that the values $P(n)$ have arbitrarily large prime factors?
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Yes suppose that all values of $P(n)$ are divisible by the prime numbers $p_1,p_2,...,p_r$ only. Now consider $P(P(0)p_1p_2...p_r)$, if $P(0)$ is different from zero we find that none of the primes $p_1,p_2,...,p_r$ divides $P(P(0)p_1p_2...p_r)/P(0)$.(a contradiction). If $P(0)=0$, then $P(0)$ is divisible by arbitrary large primes.